Extending Partial Representations of Interval Graphs

Interval graphs are intersection graphs of closed intervals of the real-line. The well-known computational problem, called recognition, asks whether an input graph G can be represented by closed intervals, i.e., whether G is an interval graph. There are several linear-time algorithms known for recognizing interval graphs, the oldest one is by Booth and Lueker (J Comput Syst Sci 13: 335-379, 1976) based on PQ-trees. In this paper, we study a generalization of recognition, called partial representation extension. The input of this problem consists of a graph G with a partial representation R' fixing the positions of some intervals. The problem asks whether it is possible to place the remaining interval and create an interval representation R of the entire graph G extending R'. We generalize the characterization of interval graphs by Fulkerson and Gross (Pac J Math 15: 835-855, 1965) to extendible partial representations. Using it, we give a linear-time algorithm for partial representation extension based on a reordering problem of PQ-trees.